Answer
3.56 cm
Work Step by Step
The molar mass of titanium is 47.90 g/mol . Thus, the mass of titanium containing\[2.55\,\times \,{{10}^{24}}\,atoms\] is calculated as:
\[\begin{align}
& 1\,\,mol\,\,\,Ti\,\,=\,\,47.90\,\,g \\
& 1\,\,mol\,\,Ti\,=\,\,6.022\,\times \,{{10}^{23}}\ \ Ti\,\,atoms \\
& Thus,\,\,\,6.022\,\times \,{{10}^{23}}\ \ Ti\,\,atoms\,\,=\,\,47.90\,\,g \\
& \,\,\Rightarrow 2.55\,\times \,{{10}^{24}}\,\,\,Ti\,\,atoms\,\,=\,47.90\,g\,\,\times \,\frac{2.55\,\times \,{{10}^{24}}\,\,Ti\,\,atoms}{\,6.022\,\times \,{{10}^{23}}\ \ Ti\,\,atoms\,\,} \\
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,202.8\,\,g \\
\end{align}\]
Calculate the volume of titanium as:
\[\begin{align}
& \,\,\,\,density\,\,=\,\,\frac{mass}{volume} \\
& \Rightarrow volume\,\,=\,\frac{mass}{density} \\
\end{align}\]
Substitute density of titanium \[=4.50\ g/c{{m}^{3}}\] and mass of titanium = 202.8 g in the above expression as:
\[volume\,\,=\,\,\frac{202.8\,\,g}{4.50\,\,g/c{{m}^{3}}}\,\,=\,45.0\ c{{m}^{3}}\]
As a titanium atom is cubical, the volume of a titanium atom is given by:
\[\begin{align}
& volume\,\,=\,\,{{\left( a \right)}^{3}}\,\,\,\,\,=45\,c{{m}^{3}} \\
& a\,\,\,\,\,\,\,\,\,\,\,=\,\,\sqrt[3]{45\,c{{m}^{3}}} \\
& a\,\,\,\,=\,\,3.556\,\,cm\,\,\approx \,3.56\,cm \\
\end{align}\]
The edge length of a titanium cube containing \[2.55\,\times \,{{10}^{24}}\,atoms\] is 3.56 cm.
